Topology of branched surfaces which admit expanding immersions

We investigate the topology of branched surfaces K which have the disjoint union of embedded circles as their branch sets S K , and which admit expanding immersions f with injective induced homomorphisms f * : π 1 ( K , * ) → π 1 ( K , f ( * ) ) . If every connected component L of K ∖ S K is orientable, then L is homeomorphic to a surface of genus ⩽1 with holes. In particular if there is a component homeomorphic to a 2-torus with holes, then K is the union of immersed tori. If every L is a 2-sphere with holes, under an additional assumption K is the union of immersed annuli.